Atlas › Modular forms › Weight 2 › Level 937 › 937.2.a

Newspace 937.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(937))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
937.2.a.a	34	degree 34	$34q - 13q^{2} - 15q^{3} + 27q^{4} - 9q^{5} - q^{6} - 18q^{7} + O(q^{8})$	+	$-1$
937.2.a.b	43	degree 43	$43q + 12q^{2} + 13q^{3} + 50q^{4} + 7q^{5} + 3q^{6} + 12q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
937.2.a.a	$x^{34} + 28x^{33} + 286x^{32} + 880x^{31} - 5870x^{30} - 53945x^{29} - 71723x^{28} + 746313x^{27} + 2949163x^{26} - 2524847x^{25} - 33583966x^{24} - 31518380x^{23} + 188809875x^{22} + 399398464x^{21} - 524119607x^{20} - 2092347719x^{19} + 262928545x^{18} + 6262134671x^{17} + 2817202187x^{16} - 11389460505x^{15} - 9343364459x^{14} + 12513387930x^{13} + 14033301360x^{12} - 7949619850x^{11} - 11403425845x^{10} + 2826125922x^{9} + 5064225359x^{8} - 703770747x^{7} - 1246429464x^{6} + 166212277x^{5} + 159149294x^{4} - 29388472x^{3} - 7483257x^{2} + 2216500x - 140788$
937.2.a.b	$x^{43} - 25x^{42} + 150x^{41} + 1278x^{40} - 18196x^{39} + 17199x^{38} + 691114x^{37} - 2772488x^{36} - 11435149x^{35} + 89681667x^{34} + 37083607x^{33} - 1513496120x^{32} + 1833736369x^{31} + 15277024166x^{30} - 36735437159x^{29} - 93693086740x^{28} + 360009734864x^{27} + 313484652418x^{26} - 2212190034325x^{25} - 175670925526x^{24} + 9182521923789x^{23} - 3289939373106x^{22} - 26667319243353x^{21} + 16001592822754x^{20} + 55353319214789x^{19} - 38539921424737x^{18} - 83282186047583x^{17} + 54838132979630x^{16} + 91039865218435x^{15} - 45474828321928x^{14} - 70441308225023x^{13} + 18774411658812x^{12} + 35628305533786x^{11} - 1385765304534x^{10} - 10184911772257x^{9} - 1164756999878x^{8} + 1484706340949x^{7} + 304690084083x^{6} - 100416646223x^{5} - 25924749294x^{4} + 2717032179x^{3} + 740607621x^{2} - 17921372x - 601316$

Orbit

Characteristic polynomial of

T

937.2.a.a

x^{34} + 28x^{33} + 286x^{32} + 880x^{31} - 5870x^{30} - 53945x^{29} - 71723x^{28} + 746313x^{27} + 2949163x^{26} - 2524847x^{25} - 33583966x^{24} - 31518380x^{23} + 188809875x^{22} + 399398464x^{21} - 524119607x^{20} - 2092347719x^{19} + 262928545x^{18} + 6262134671x^{17} + 2817202187x^{16} - 11389460505x^{15} - 9343364459x^{14} + 12513387930x^{13} + 14033301360x^{12} - 7949619850x^{11} - 11403425845x^{10} + 2826125922x^{9} + 5064225359x^{8} - 703770747x^{7} - 1246429464x^{6} + 166212277x^{5} + 159149294x^{4} - 29388472x^{3} - 7483257x^{2} + 2216500x - 140788

937.2.a.b

x^{43} - 25x^{42} + 150x^{41} + 1278x^{40} - 18196x^{39} + 17199x^{38} + 691114x^{37} - 2772488x^{36} - 11435149x^{35} + 89681667x^{34} + 37083607x^{33} - 1513496120x^{32} + 1833736369x^{31} + 15277024166x^{30} - 36735437159x^{29} - 93693086740x^{28} + 360009734864x^{27} + 313484652418x^{26} - 2212190034325x^{25} - 175670925526x^{24} + 9182521923789x^{23} - 3289939373106x^{22} - 26667319243353x^{21} + 16001592822754x^{20} + 55353319214789x^{19} - 38539921424737x^{18} - 83282186047583x^{17} + 54838132979630x^{16} + 91039865218435x^{15} - 45474828321928x^{14} - 70441308225023x^{13} + 18774411658812x^{12} + 35628305533786x^{11} - 1385765304534x^{10} - 10184911772257x^{9} - 1164756999878x^{8} + 1484706340949x^{7} + 304690084083x^{6} - 100416646223x^{5} - 25924749294x^{4} + 2717032179x^{3} + 740607621x^{2} - 17921372x - 601316

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(937))$	77
Old forms	0
Cusp forms $S_{2}(\Gamma_0(937))$	77
Eisenstein series $E_{2}(\Gamma_0(937))$	1
Modular forms $M_{2}(\Gamma_0(937))$	78

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.303 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 937]: [0, 78] (certified at ell = 2147472289)
T = [(2, 1), (3, 1)]; h of degree 77 = 1 x 77 (phi(ord eps) x dim S_k^new); 8 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(937, 2)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 937:2 1000

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